Geodesic
Critical Bellman--Harris branching processes with long-living particles
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 257-287

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A critical indecomposable two-type Bellman–Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter $\beta\in(0,1]$. It is shown that, contrary to the critical indecomposable Bellman–Harris branching processes with finite variances of the life-lengths of particles of both types, the probability of observing first-type particles at a distant moment $t$ is infinitesimally less than the survival probability of the whole process. In addition, a Yaglom-type limit theorem is proved for the distribution of the number of the first-type particles at moment $t$ given that the population contains particles of the first type at this moment. 
@article{TM_2013_282_a18,
     author = {V. A. Vatutin and V. A. Topchii},
     title = {Critical {Bellman--Harris} branching processes with long-living particles},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {257--287},
     publisher = {mathdoc},
     volume = {282},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_282_a18/}
}
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V. A. Vatutin; V. A. Topchii. Critical Bellman--Harris branching processes with long-living particles. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 257-287. http://geodesic.mathdoc.fr/item/TM_2013_282_a18/